journal of harmonized research in engineering 2(2), 2014 ... · by the theory of elasticity methods...

Brovman M., J. Harmoniz. Res. Eng. 2014, 2(2), 251-258

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For Correspondence: brovmanATmail.ru Received on: March 2014 Accepted after revision: April 2014 Downloaded from: www.johronline.com

ISSN 2347 – 7393

Introduction Problems related to symmetrical deformation of the sphere with constant stuff density were considered in studies [1, 2]. They were tackled by the theory of elasticity methods provided that mechanical properties of the sphere stuff and its density were known. The problem factored in density variation was solved by another study [3]. If a solid sphere is compressed under the prohibition of mass loss, its density increases causing a structural strain of a body. The

external pressure enhances gravitation, therefore gravitation and pressure deformation values cannot be summed up as it is customary to do in a linear elastic theory. Hence, there occurs a possibility of sphere buckling. 1. Sphere Buckling To consider buckling of the sphere subjected to gravitation and external pressure see Fig.1. A small body of dimensions and mass

is effected by radial gravitation force

Where φ, r – are spherical system coordinates, m2 – mass of sphere with radius r, δ – gravitation constant.

Abstract: Deformation of solid bodies under gravitation forces is of frequent occurrence and a subject matter of Mechanics. Usually calculations are performed for sphere-shaped solid bodies of a constant density. But if a sphere is compressed whilst its mass remains constant, a density is to increase in the process of deformation. Under these circumstances buckling and even collapse of a sphere are found to occur. Rotation of sphere-shaped solid bodies is also taken into consideration. Keywords: elastic deformation; density increase; buckling; unconfined compression of solid bodies.

Journal Of Harmonized Research in Engineering 2(2), 2014, 251-258

GRAVITATIONAL DEFORMATION OF SOLID BODIES

Mikhail Brovman

Chief Scientist, Osvastka

Original Research Article

Journal Of Harmonized Research (JOHR)



where r1 – additional argument, ρ(r1) – variable density under stuff compression. By Newton's law the static equilibrium equation is

(1)

where – components of stress in a

spherical coordinate system (the third main stress component is equal to ).

Mean strain is determined by value

where , are components of strain; – radial displacement of body points under strain.

Density is equal to

and as it is shown in [3] it can change under deformation. Given that density increases under compression but does not depend on coordinate r and sphere surface displacement is equal to

(R – initial sphere radius without deformation, U0 – non-dimensional value), the following formula can be developed:

(2)

where – initial stuff density; value describes a total change of sphere density.

Similar to [3], to solve the equation (1) is possible with the help of Hooke's law:

(3)

,

(4)

where E – stuff elasticity modulus, γ – the Poisson ratio. It is convenient to use two non-dimensional parameters: a and b

(5)

(6)

The first parameter (a) defines the relationship between gravitation and material rigidity, while the second parameter (b) – between external pressure Ρ and material rigidity. For the surface of sphere with Ur= -U0R and r = R(1-U0) it is possible to establish the following equation from (3):

(7) 2. Numerical Calculations Fig.2 represents functions when b = 0; 0.25; 0.50 and 1.0. If external pressure at the sphere surface is equal to zero, a lower curve is . At point A with a = 0.105, U0 = 0.25

the derivative is infinite. Point A

becomes an unstable point. The upper part of this curve cannot describe a real body deformation (it is represented by thin line in Fig.2). If to assume that density remains constant and equal to ρ0, the following equation can be obtained: . It is represented by the dashed line in Fig.2. It may be useful only if

( see Fig.2). Under the very circumstances, gravitation and external pressure displacement values may be summed up.



So, there may be seen an upper limit of sphere dimension with radius R, under the given density ρ0. If 0.105, it is possible to derive from (5) a critical radius

(8)

Critical mass of a solid sphere corresponds to the value

(9)

For example, if there is a solid stuff with

;

;

the value of makes

,

and

The equilibrium of such a sphere or larger in size spheres is unfeasible. When its density reaches 2.37 , a gravitational collapse occurs. Two assumptions are to be taken into consideration: 1. Hooke's law is valid and the elasticity

modulus remains constant. 2. Newton’s law of gravitation is an established

truth.

Reinforcement of elasticity modulus was considered in article [3]. If its change is not considerable, the main conclusions remain valid. Plastic deformation was also discussed in [3]. It was proved that plastic deformation cannot invalidate the main conclusions [see 3, 4]. In case of gravitation collapse, when a gravity force is too strong the general theory of relativity laws are to be applied instead of the classical theory and Newton’s law. According to external-reference coordinate system when a collapse takes place and the radius approaches to a gravity radial value, all the body processes become slow [5].

where C is the velocity of light in

vacuum. In our example ;

;

. As may be inferred from the example given below, the relation between critical radius

and gravitational radius is equal, so the

Newton's law can be applied.

and .

Buckling and collapse will take place right before the necessity to apply the theory of relativity arises. It is well known that if a stellar mass is twice or thrice as much as the Sun mass, the gravitation collapse occurs and the star becomes a black hole [5]. Such process is eventual not only for plasma but for a solid body with definite characteristics. 3. Artificial Collapse Potential The possibility of artificial buckling followed by a body collapse is very important nowadays. But this article does not aim at discussing the consequences of potential collapse in relation to the Earth (it is the subject of the other paper).



We would like to estimate the probability of such a phenomenon as itself. Based on equation (7), it is high. To produce a collapse and, hence, an artificial black hole it is sufficient to create the solid body surface pressure. From equation (7) it can be inferred that if a surface of the sphere with is

subjected to external pressure with , then

at (10)

This value corresponds to a critical one:

(11)

Critical radius and critical mass are equal:

,

(12)

where and are critical values of

and , provided that external pressure is

equal to zero and . For example, if one wants to produce a collapse with the stuff mentioned above, namely:

;

;

but it is

necessary to reduce from

(when ) by times.

It is possible if ;

.

If ; the applied

pressure is calculated by formula (6):

In circumstances where the pressure

is very high mechanical properties can vary considerably so one is to be

careful in drawing conclusions. But now we cannot be absolutely certain that such a phenomenon as buckling of a solid sphere is impossible. It is known that a solid sphere has natural oscillation frequencies.

; …, where is the

sound velocity,

So, first two frequencies are ;

.

For

;

But under compression the values R and change, therefore resonance frequencies turn into off-resonance frequencies. Under compression R changes into and

into and

(13)

If one produces the external pressure oscillation on the sphere surface and changes its frequency by modifying in accordance with equation (13), it becomes possible to incite resonance and collapse. Sphere oscillation dynamics is intended to be treated at length elsewhere. Naturally, to do this it is necessary to carry out continuous measurements of displacement and value. The calculations of plastic deformation are considered in [3]. It is to take place on the sphere surface conditioned upon:

where is a yield stress of the



stuff (see formula (4)). The calculations suggest that these are the very conditions for buckling phenomenon to occur. Change of volume and mean stress are proportional even in plastic deformation, therefore data on plasticity are compatible with main conclusions. The calculations for large deformations were performed with the help of Cauchy-Green strain tensor and tensor-gradient in [3]. Instead of equation (7) the calculation for John’s semi-linear stuff suggests the following [3]:

(14) The analysis of this equation shows that buckling occurs when

and if , then

Though critical value differs from formula

(7), the main conclusion of limiting values

and is fully justified. 4. Rotation of Solid Sphere If the sphere rotates about axis Ƶ (see Fig.3) with angular velocity , every mass experiences not only a gravitational force but a centrifugal one:

This force acts straight across axis Ƶ. Fig.3 shows the initial configuration of sphere with radius as well as its shape deformed under the forces of gravitation and inertia. Under the circumstances the body is not a sphere anymore and its Ƶ- axis displacement is not equal to

displacement in -direction (Fig.3). We shall call a sphere surface displacement in Ƶ-direction – and a sphere surface

displacement in the straight across axis Ƶ direction – .

Obviously, when a sphere rotates:

and even , so configuration of a rotating sphere is close to ellipsoid with the volume:

.

Deformation of a rotating sphere with constant values and was studied in [1]. But here we suppress previous assumption and the equations for the two displacement values and will be:

(15)

where and from equations (5, 6) go with additional non-dimensional parameter:

(16)

under initial values of density and angular velocity – and This parameter characterizes correlation between centrifugal forces and rigidity of sphere stuff. Values and are functions of Poisson

ratio .



(17)

If , , then instead of two equations (15) we get one (7). But if and it is necessary to take into account that body mass and its angular momentum remain constant. Therefore, the sphere density in compression increases while the moment of inertia Ј decreases. Angular velocity cannot remain constant as it has to increase in order to ensure the stability of angular momentum i.e. value Ј . Thence we obtain:

(18)

(19)

If there is neither loss of stuff nor action of external forces, a mass and an angular momentum are to remain constant. Momentum is equal to Ј = Ј0 where Ј – moment of inertia and Ј0 – its nominal value. Fig.4 illustrates the calculation results of equation (15), for ; ;

when and .

We can see that if

( ), buckling phenomenon takes place.

If , equilibrium state is impossible. Judging from upper parts of the curves, compression in z-axis direction gives

and zero thickness. But

and induce a compression not to a point (black hole) but to a one-dimensional disk body with radius and zero thickness.

Fig.5 graphs and are made for

( ). Buckling takes place

when . In that

case ; so there be

formed a disk of zero thickness with R=1.3 ( ). If , centrifugal forces are strong and collapse sphere diameter increases straight across axis Ƶ (Fig.3). Graphs of functions , will correspond to those of Fig.6 if angular velocity increases, for instance up to . With the

increase of and , the stability region of a

solid body degrades. If , buckling and collapse occur when . In

the vicinity of a change is immediate and one can observe an acute angle in Fig.6a. To demonstrate a rapid



change of function near value

a part of this curve near

point , is scaled up

(Fig.6b). Calculations suggest, when there results a limiting case: the equilibrium is impossible and the stuff rigidity cannot resist its buckling. But there exists an angular velocity

value which is

equal to

when sphere density ,

radius and modulus of elasticity

.

Post-buckling equilibrium is impossible and it is a problem of dynamic deformation (it is the subject of the other paper). But after the buckling takes place two variants are possible: rapture of sphere with its parts flying apart; and collapse with turning into the object of zero thickness and radius . Mass

( and momentum (Ј0 to be sure,

remain constant. Such an object may be called ‘black disk’. Certainly, the first variant is more likely to occur but one cannot exclude the other alternative. Both variants of dynamic deformation result in explosion. Conclusions 1. Existence of a sphere-shaped sold body of

arbitrary large mass is impossible. There is a limiting value of mass when gravitation force produces buckling and unconfined compression to the point.

2. Solid body buckling can be produced artificially by pressure exerted on the sphere surface. The formulas in question are presented.

3. If a solid sphere rotates, its deformation under gravitation and inertia can result in buckling but without a central symmetry, which causes the transformation of a sphere into a disk of zero thickness and infinite density. Therefore, both the possible mass of a solid sphere and its permissible angular velocity of rotation are limited.

References 1. Sedov L.ed.1994.Mechanics of continuous

matter, Moscow, Russia.Science.601.ISBN 5-02-007052-1.



2. Iliushin A.ed.2004.Plasticity,Moscow, Russia. Moscow University,376.ISBN 5-94010-377-4.

3. Brovman M.Ja.Elastoplastic Deformation of a Sphere by Symmetrical Loading.Mechanic of Solid State.1994.3.117-125.

4. Brovman M.Ja.Deformation of Sphere by Symmetrical Loading. Ingenerno-physical Journal.1999.Vol.72.5.1047-1048.

5. Rubin S. ed.2006.Structure of our universe. Frjasino. Russia. Vek 2. 312.ISBN 585099-166-2.

journal of harmonized research in engineering 2(2), 2014 ... · by the theory of elasticity methods...

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